PAijpam.eu SOLVING INTUITIONISTIC FUZZY MULTI-OBJECTIVE LINEAR PROGRAMMING PROBLEMS USING RANKING FUNCTION

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1 International Journal of Pure and Applied Mathematics Volume 106 No , ISSN: (printed version); ISSN: (on-line version) url: doi: /ijpam.v106i8.18 PAijpam.eu SOLVING INTUITIONISTIC FUZZY MULTI-OBJECTIVE LINEAR PROGRAMMING PROBLEMS USING RANKING FUNCTION R. Sophia Porchelvi 1, S. Uma 2 1 PG and Research Department of Mathematics A.D.M. College for Women (Autonomous) Nagapattinam, , Tamil Nadu, INDIA 2 Department of Mathematics AVC College of Engineering Mayiladuthurai, Tamil Nadu, INDIA Abstract: The concept of Fuzzy Numbers has been enhanced in many decision making problems of engineering optimization. Ranking of Fuzzy Numbers is one of the techniques that conceptualize Fuzzy Numbers to demonstrate the order of preference in decision making. This paper focuses on modified Maleki s and Yager s ranking functions on trapezoidal intuitionistic fuzzy numbers (TIFNS) to solve multi-objective intuitionistic fuzzy linear programming problem (MOIFLPP) in which both the coefficients of objective functions as well as the right-hand side are trapezoidal intuitionistic fuzzy numbers. Finally some illustrative examples for different cases with various states are given to check the effectiveness of the modified ranking functions. Key Words: trapezoidal intuitionistic fuzzy number (TIFN), modified Maleki s ranking function, modified Yager s ranking function 1. Introduction A lot of application problems can be modeled as mathematical problem which may be formulated with uncertainty. The concept of fuzzy multi-objective Received: February 15, 2016 Published: July 23, 2016 c 2016 Academic Publications, Ltd. url:

2 150 R.S. Porchelvi, S. Uma linear programming was first proposed by Zimmerann[6]. Maleki [4] introduced fuzzy variables in linear programming problems and proposed a new method for solving these problems using ranking function.in this paper, we solving the intuitionistic fuzzy multi-objective linear programming problems, when the coefficients of two objective functions are intuitionistic fuzzy numbers, as well as right-hand side are intuitionistic fuzzy numbers too, and both the coefficients of objective function as well as right-hand side are intuitionistic fuzzy numbers by using Maleki linear ranking function when (α = β) and α β and using Yager linear ranking function when (α = β) and α β. This paper is outlined as follows. In Section 2, we presented trapezoidal intuitionistic fuzzy numbers. In Section 3, we interested in ranking functions, but in Section 4, we study intuitionistic fuzzy multi-objective linear programming problems with fuzzy coefficients objective function and intuitionistic fuzzy right-hand side, then both of them. In Section 5, we illustrate all type of intuitionstic fuzzy multi-objective linear programming problems in numerical examples. Finally in Section 6 we make conclusion for this study. 2. Preliminaries Definition 1. [1] Given a fixed setx = x 1,x 2,...,x n, an intuitionistic fuzzy set (IFS) is defined as ÃI = ( x i,µãi(x i ),ϑãi(x i ) /x i X) which assigns to each element x i, a membership degree µãi(x i ) and a non-membership degree µãi(x i ) under the condition 0 µãi(x i )+νãi(x i ) 1, for all x i X Definition 2 (Trapezoidal Intuitionistic Fuzzy Numbers). There are various types of fuzzy numbers,in which the triangular and trapezoidal are the most important fuzzy memperships. In this research we use the trapezoidal intuitionistic fuzzy numbers. In fact the fuzzy number is defined by its corresponding membership function and non membership function as follows: { x a1 µã(x) =,when a 1 x a 2,µã(x) = 1, when a 2 x a 3 a 2 a 1 { a4 x µã(x) =,when a 3 x a 4,µã(x) = 0, otherwise a 4 a 3 { a2 x νã(x) =,when a 1 x a 2,νã(x) = 0, when a 2 x a 3 a 2 a 1 { x a3 νã(x) = when a 3 x a 4,νã(x) = 1, otherwise a 4 a 3

3 SOLVING INTUITIONISTIC FUZZY MULTI-OBJECTIVE Ranking Function The ranking function is the approach of ordering fuzzy numbers. Various types of ranking function have been introduced which are used to solve multi objective linear programming problems with fuzzy parameters. The ranking function of F(R) is as follows: ã b then R(ã) R( b), if ã > b then R(ã) > R( b), if ã = b then R(ã) = R( b) 3.1. Maleki Ranking Function ã = (a l,a u,α,β),be a fuzzy numbers, then the ranking function is R(ã) = 1 0 (infã λ +supã λ ) dλ when α = β R(ã) = a l +a u + 1 (β α) when α β Yager Ranking Function ã = (a l,a u,α,β),be a fuzzy numbers, then the ranking function is R(ã) = a l αl l (λ)dλ+ a u βr l (λ)dλ, when α = β 2 R(ã) = [a l +a u 45 α+ 23 β ], when α β 0 4. Intuitionistic Fuzzy Multi-Objective Linear Programming Problem The crisp multiobjective linear programming problem is defined as follows: maxz 1 = c j x j,maxz 2 = c k x k, j 1 k a ij x j b i. j 1 k 1

4 152 R.S. Porchelvi, S. Uma In this paper we study three states of intuitionistic fuzzy multi objective linear programming problem. The first state are intuitionistic fuzzy numbers in the objective function coefficients,the second state are intuitionistic fuzzy numbers in the right hand side coefficients, the third state are intuitionistic fuzzy numbers for both in the objective function coefficients and the right hand side coefficients.we have formulated the above three fuzzy multi objective linear programming problem states, as follows: First State. In this section the objective function coefficients as trapezoidal intuitionistic fuzzy numbers which as maxz 1 = c j x j,maxz 2 = c k x k k=1 ã ij x j b i where c j fuzzy coefficients of objective function, then we solve the fuzzy multiobjective linear programming by using two ranking function: Maleki ranking function: maxz = [c lj +cuj + 12 (β α) ] x j, a ij x j b i. Yager ranking function: maxz = j 1 1 [c lj +c uj 45 2 α+ 23 ] β x j, a ij x j b i. Second State. In this state, we make the right hand side coefficients as intuitionistic trapezoidal fuzzy numbers which as maxz 1 = c j x j,maxz 2 = c k x k k=1 a ij x j b i. Multi objective linear programming problem when using maleki ranking function: a ij x j [b lj +buj + 12 ] (β α)

5 SOLVING INTUITIONISTIC FUZZY MULTI-OBJECTIVE Multi objective linear programming problem when using Yager ranking function: maxz 1 = c j x j, maxz 2 = c k x k, k=1 a ij 1 2 [b lj +b uj 45 α+ 23 β ]. Third State. In this state, we make both the objective function coefficients and right hand side as trapezoidal intuitionistic fuzzy numbers which as maxz 1 = c j x j,maxz 2 = c k x k k=1 ã ij x j b i Multi objective linear programming problem when using Maleki ranking function a ij 1 2 [b lj +buj 45 α+ 23 β ] x i, a ij 1 2 [b lj +buj 45 α+ 23 β ]. Multi objective linear programming problem when using Yager ranking function maxz = 1 [c lj 2 +cuj 45 α+ 23 ] β x j, a ij 1 2 [b lj +buj 45 α+ 23 β ].

6 154 R.S. Porchelvi, S. Uma 5. Numerical Examples In this section we discussed all states by the following examples which is suggested by the researchers in case of α = β or α β. Example 1. The following example is suggested by the researchers in case of α = β. (a) max Z 1 = (4,2,3,3)(1,5,2,2) x 1 +(1,3,2,2)(3,1,2,2) x 2 +(1,1,3,3)(1,1,2,2) x 3, max Z 2 = (1,2,2,2)(2,1,3,3) x 1 +(3,2,2,2)(4,1,2,2) x 2 +(2,2,3,3)(3,1,2,2) x 3 2 x 1 x 2 +2 x 3 4, x 1 +4 x 3 4, x 1 +3 x 2 +2 x 3 7, x 1, x 2, x 3 0 By using Maleki ranking function we get maxz 1 = 6x 1 +4x 2 +2x 3, maxz 2 = 3x 1 +5x 2 +4x 3, 2x 1 x 2 +2x 3 4,x 1 +4x 3 4,x 1 +3x 2 +2x 3 7 We solve the above crisp linear programming by using through TORA software program we get the following solution, x 1 = 2.71,x 2 = 1.43, x 3 = 0,z = 22. maxz 2 = 3x 1 +5x 2 +4x 3, 2x 1 x 2 +2x 3 4,x 1 +4x 3 4, x 1 +3x 2 +2x 3 7,2.71x x 2 22 By using simplex method, the solution is x 1 = 2.71,x 2 = 1.43,x 3 = 0,z = If each stage of preemptive optimization yields a single objective optimum, the final solution is an efficient point of the full multi-objective model. By using Yager ranking function we get maxz 1 = 2.8x x x 3, 2x 1 x 2 +2x 3 4,x 1 +4x 3 4,x 1 +3x 2 +2x 3 7 by using simplex method the solution is x 1 = 2.71,x 2 = 1.43,x 3 = 0,z =

7 SOLVING INTUITIONISTIC FUZZY MULTI-OBJECTIVE maxz 2 = 1.37x x x 3, 2x 1 x 2 +2x 3 4,x 1 +4x 3 4, x 1 +3x 2 +2x 3 7,2.71x x On solving, the solutions is x 1 = 2.71,x 2 = 1.43,x 3 = 0,z = (b) max Z 1 = 4 x 1 +3 x 2 +5 x 3, max = 3 x 1 +5 x 2 +9 x 3, 2x 1 x 2 +2x 3 (2,6,3,3)(3,5,2,2),x 1 +4x 3 (2,7,2,2),(4,5,3,3), x 1 +3x 2 +2x 3 (5,9,3,3)(6,8,2,2) By using Maleki ranking function we get 2x 1 x 2 +2x 3 8,x 1 +4x 3 9,x 1 +3x 2 +2x 3 14 maxz 1 = 4x 1 +3x 2 +5x 3, 2x 1 x 2 +2x 3 8,x 1 +4x 3 9,x 1 +3x 2 +2x 3 14 By using simplex method the solution is x 1 = 5.43,x 2 = 2.86,x 3 = 0,z = maxz 2 = 3x 1 +5x 2 +9x 3, 2x 1 x 2 +2x 3 8,x 1 +4x 3 9, x 1 +3x 2 +2x 3 14,5.43x x The solution is x 1 = 0,x 2 = 3.17,x 3 = 2.25,z = If each stage of preemptive optimization yields a single objective optimum, the final solution is an efficient point of the full multi-objective model. By using Yager ranking function we get, maxz 1 = 4x 1 +3x 2 +5x 3, 2x 1 x 2 +2x 3 3.8,x 1 +4x 3 3.3,x 1 +3x 2 +2x by using simplex method, the solution is, x 1 = 2.60,x 2 = 1.40,x 3 = 0,z = maxz 2 = 3x 1 +5x 2 +9x 3,

8 156 R.S. Porchelvi, S. Uma 2x 1 x 2 +2x 3 3.8,x 1 +4x 3 3.3, x 1 +3x 2 +2x 3 6.8,2.60x x byusingsimplexmethodthesolution is, x 1 = 0,x 2 = 1.72,x 3 = 0.83,z = (c) max Z 1 = (4,2,3,3)(1,5,2,2) x 1 +(1,3,2,2)(3,1,2,2) x 2 +(1,1,3,3)(1,1,2,2) x 3, max Z 2 = (1,2,2,2)(2,1,3,3) x 1 +(3,2,2,2)(4,1,2,2) x 2 +(2,2,3,3)(3,1,2,2) x 3 2x 1 x 2 +2x 3 (2,6,3,3)(3,5,2,2),x 1 +4x 3 (2,7,2,2),(4,5,3,3), x 1 +3x 2 +2x 3 (5,9,3,3)(6,8,2,2), x 1, x 2, x 3 0 By using Maleki ranking function maxz 1 = 6x 1 +4x 2 +2x 3,maxz 2 = 3x 1 +5x 2 +4x 3 2x 1 x 2 +2x 3 8,x 1 +4x 3 9,x 1 +3x 2 +2x 3 14 by using simplex method the solution is x 1 = 5.43,x 2 = 2.86,x 3 = 0,z = 44. maxz 2 = 3x 1 +5x 2 +4x 3 2x 1 x 2 +2x 3 8,x 1 +4x 3 9, x 1 +3x 2 +2x 3 14,5.43x x 2 44 by using simplex method x 1 = 5.43,x 2 = 2.86,x 3 = 0,z = By using Yager ranking function we get maxz 1 = 2.8x x x 3 2x 1 x 2 +2x 3 3.8,x 1 +4x 3 3.3,x 1 +3x 2 +2x by using simplex method x 1 = 2.60,x 2 = 1.40,x 3 = 0,z = maxz 2 = 1.37x x x 3, 2x 1 x 2 +2x 3 3.8,x 1 +4x 3 3.3,

9 SOLVING INTUITIONISTIC FUZZY MULTI-OBJECTIVE x 1 +3x 2 +2x ,60x x the solution is x 1 = 2.60,x 2 = 1.40,x 3 = 0,z = Example 2. The following example is suggested by the researchers in case of α β. (a) max Z 1 = (4,2,3,1)(5,1,2,3) x 1 +(1,3,2,1)(2,2,3,1) x 2 +(1,1,2,3)(1,1,3,2) x 3 max = (1,2,1,2)(2,1,3,1) x 1 +(3,2,1,3)(4,1,2,1) x 2 +(2,2,1,3)(3,1,3,2) x 3 2 x 1 x 2 +2 x 3 4, x 1 +4 x 3 4, x 1 +3 x 2 +2 x 3 7, x 1, x 2, x 3 0 Solution : By using Maleki ranking function we get maxz 1 = 5x x x 3,maxz 2 = 3.5x 1 +6x 2 +5x 3 2x 1 x 2 +2x 3 4,x 1 +4x 3 4,x 1 +3x 2 +2x 3 7 The solution is x 1 = 2.71,x 2 = 1.43,x 3 = 0,z = maxz 2 = 3.5x 1 +6x 2 +5x 3 2x 1 x 2 +2x 3 4,x 1 +4x 3 4, x 1 +3x 2 +2x 3 7,2.71x x The solution is x 1 = 2.71,x 2 = 1.43,x 3 = 0,z = By using Yager ranking function we get maxz 1 = 2.13x x x 3, 2x 1 x 2 +2x 3 4,x 1 +4x 3 4,x 1 +3x 2 +2x 3 7 The solution is x 1 = 2.71,x 2 = 1.43,x 3 = 0,z = maxz 2 = 1.76x x x 3 2x 1 x 2 +2x 3 4,x 1 +4x 3 4,

10 158 R.S. Porchelvi, S. Uma x 1 +3x 2 +2x 3 7,2.71x x The solution is x 1 = 2.25,x 2 = 1.31,x 3 = 0.4,z = (b) max Z 1 = 4 x 1 +3 x 2 +5 x 3, max = 3 x 1 +5 x 2 +9 x 3 2x 1 x 2 +2x 3 (2,6,1,3)(3,5,2,3),x 1 +4x 3 (2,7,2,3)(1,8,3,2), x 1 +3x 2 +2x 3 (5,9,1,2)(6,8,2,3) By using Maleki ranking function we get Subject to maxz 1 = 4x 1 +3x 2 +5x 3 2x 1 x 2 +2x 3 9,x 1 +4x 3 9.5,x 1 +3x 2 +2x The solution is x 1 = 5.93,x 2 = 2.86,x 3 = 0,z = maxz 2 = 3x 1 +5x 2 +9x 3 2x 1 x 2 +2x 3 9,x 1 +4x 3 9.5, x 1 +3x 2 +2x ,5.93x x The solution is x 1 = 0,x 2 = 3.25,x 3 = 2.38,z = If each stage of preemptive optimization yields a single objective optimum, the final solution is an efficient point of the full multi-objective model. By using Yager ranking function we get maxz 1 = 4x 1 +3x 2 +5x 3, 2x 1 x 2 +2x 3 4.6,x 1 +4x 3 4.7,x 1 +3x 2 +2x 3 by using simplex method the solution is x 1 = 3.01,x 2 = 1.42,x 3 = 0,z = maxz 2 = 3x 1 +5x 2 +9x 3 2x 1 x 2 +2x 3 4.6,x 1 +4x 3 4.7, x 1 +3x 2 +2x ,3.01x x The solutions are x 1 = 0,x 2 = 1.64,x 3 = 1.18,z = (a) max Z 1 = (4,2,3,1)(5,1,2,3) x 1 +(1,3,2,1)(2,2,3,1) x 2

11 SOLVING INTUITIONISTIC FUZZY MULTI-OBJECTIVE (1,1,2,3)(1,1,3,2) x 3 max Z 2 = (1,2,1,2)(2,1,3,1) x 1 +(3,2,1,3)(4,1,2,1) x 2 +(2,2,1,3)(3,1,3,2) 2x 1 x 2 +2x 3 (2,6,1,3)(3,5,2,3),x 1 +4x 3 (2,7,2,3)(1,8,3,2), x 1 +3x 2 +2x 3 (5,9,1,2)(6,8,2,3), x 1, x 2, x 3 0 By using Maleki ranking function maxz 1 = 5x x x 3,maxz 2 = 3.5x 1 +6x 2 +5x 3, 2x 1 x 2 +2x 3 9,x 1 +4x 3 9.5,x 1 +3x 2 +2x 3 1 The solution is x 1 = 5.93,x 2 = 2.86,x 3 = 0,z = maxz 2 = 3.5x 1 +6x 2 +5x 3, 2x 1 x 2 +2x 3 9,x 1 +4x 3 9.5, x 1 +3x 2 +2x ,5.43x x The solution is x 1 = 4.53,x 2 = 2.51,x 3 = 1.23,z = By using Yager ranking function we get maxz 1 = 2.13x x x 3 2x 1 x 2 +2x 3 4.6,x 1 +4x 3 4.7,x 1 +3x 2 +2x The solution is x 1 = 3.01,x 2 = 1.42,x 3 = 0,z = maxz 2 = 1.76x x x 3 2x 1 x 2 +2x 3 4.6,x 1 +4x 3 4.7, x 1 +3x 2 +2x ,3.01x x The solution is x 1 = 2.26,x 2 = 1.26,x 3 = 0.61,z = 9.47.

12 160 R.S. Porchelvi, S. Uma 6. Conclusion The coefficients of the objective function and the right-hand side with intuitionstic fuzzy numbers are ranked with two special ranking functions for Maleki and Yager linear ranking function. For all twelve states which is studied in the paper, with α = β and α β. When the objective function coefficients are intuitionstic fuzzy numbers the preferable solution is x 1 = 2.71,x 2 = 1.43,x 3 = 0,z = When the right hand side coefficients are intuitionstic fuzzy numbers the preferable solution is x 1 = 0,x 2 = 3.25,x 3 = 2.38,z = When the objective function coefficients and right hand side coefficients are intuitionstic fuzzy numbers the preferable solution is x 1 = 4.53,x 2 = 2.51,x 3 = 1.23,z = References [1] Atanassov.K.T (1986), Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, [2] Atanassov.K.T (1989), More on intuitionistic fuzzy sets, Fuzzy sets and systems, 33, [3] Bellman,R.E. and Zadeh, L.A., Decision making in a fuzzy environment, Management Science, 17, ,1970. [4] Maleki, H.R. 2002, Ranking functions and their applications to fuzzy linear programming, Far East Journal Mathematics Sciences, 4(2002),pp: [5] Yager,R,R.1981, A procedure for ordering fuzzy subsets of the unit interval, Information Sciences, vol.24, no.2, pp: [6] Zimmerman H. J., Fuzzy Programming and Linear Programming with Several Objective Functions, Fuzzy Sets and Systems, Vol. 1, 1978,